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tax even the most powerful of our modern computers. If these calculations could be accomplished, however, the advantage of a direct calculation of turbulence would be that no approximations or empirical postulates are required.

Ensemble Averages. Largely for the reasons given above, almost all theoretical approaches to turbulence modeling use some type of averaging either temporal, spatial, or ensemble. With the proper statistical treatment, the solution of turbulent flow problems need not resolve the full spectrum of eddies, initial and boundary conditions need not be specified in minute detail, and a flow whose mean velocity is one-dimensional can be numerically calculated in one dimension even though the resolved turbulence is three-dimensional. However, with these advantages for turbulence transport modeling come the disadvantages of assumptions and approximations needed to obtain a set of solvable equations.

What is meant by the average of any flow variable in a turbulent flow? Time averages are easy to understand. We say that fluid flow is statistically steady if the time average of many fluctuations at some point in space is independent of the averaging period chosen. Spatial averages, likewise, are easy to visualize but are relevant only when the structural scale of the turbulence is very small compared with that of the mean-flow fluctuations—a relatively rare condition. Here we will focus on ensemble averaging, which is the most general type of averaging with the fewest restrictions.

We can intuitively sense what an ensemble average is if we imagine a very large number of experiments, all with the same macroscopic initial and boundary conditions, but each with its own particular realization of the turbulent part of the flow (Fig 12). The ensemble average of some flow parameter at any given point and time is then the

average of that parameter over all the experiments. For the condition of steady flow, time and ensemble averages are the same.

The ensemble average is most conveniently formulated in terms of moments of an appropriate distribution function. (Here, rather than integral nomenclature, we will simply use an overline to designate an ensemble average.) Thus ō is the moment, or ensemble average, of the pressure (per unit density), and pu¡ is the ensemble average of the product of pressure and the ith component of fluid velocity. (Note that pu; does not necessarily equal pu¡.)

For each experiment in a series, detailed measurements give p and u;, both of which fluctuate strongly as a function of position and time. Likewise, ō and ū¡ vary with position and time but in a much calmer fashion. The difference between the individual experimental value and the ensemble average is the fluctuating part of the variable, denoted by a prime: p' = p-p; u = u; - u. The ensemble average of this fluctuating part must be zero for each variable (that is, p' = 0 and u = 0), but it does not follow that the moment, or ensemble average, of a product of fluctuational variables (such as p'u¦ or u¦u) vanishes. Indeed, the essence of our turbulence modeling is contained in the behavior of such ensemble averages of fluctuational products.

Reynolds-Stress Transport Equation. One of these fluctuational products, the Reynolds stress tensor, is especially important; it is defined by

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Notice that the contraction of the Reynolds stress tensor (that is, when i = j) is exactly twice the turbulence kinetic energy per unit mass of fluid (R;; = u{u} = 2K ).

The importance of R1; in turbulence modeling can be demonstrated quite handily. First we rewrite the Navier-Stokes equations, expressing each of the variables as the sum of its mean and fluctuating parts:

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Then we take the ensemble average of these equations (commuting averages and derivatives where necessary and remembering that the average of a single fluctuating variable is zero) and obtain the mean-flow equations:

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A single term involving the Reynolds stress has emerged, and we see that the only effect of turbulence on the mean flow is through the addition of that term to the equations. We note in passing that Eqs. 13 form the basis of point-functional turbulence models (the middle branch in Fig. 11) and will return to this point shortly.

The mean-flow equations (Eqs. 13) can be subtracted from the full equations (Eqs. 12) to show that the fluctuating parts of the variables obey the equations

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We need Eqs. 14 to derive the Reynolds-stress transport equation, that is, a description of the behavior of the Reynolds stress itself (the right branch of Fig. 11). This derivation is straightforward but tedious. We merely note that the following steps are involved:

1. multiply Eq. 14b by u to obtain Eq. 14c,

2. interchange i and j in Eq. 14c to obtain Eq. 14d,

3. add Eqs. 14c and 14d, and

4. take the ensemble average.

With some rearrangement of terms and the identification of R1; in several places, the result is

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We now can distinguish the turbulence-transport theories and their predecessors, the point-functional turbulence theories. As we remarked earlier, point-functional turbulence theories use Eqs. 13 by postulating a form for the Reynolds stress R1; that is a function of the mean-flow variables themselves. As a result, such theories are called "point-functional" because the description of the turbulence at some point in the flow depends only on the current value of the mean-flow variables. Point-functional theories have the advantage of being as easy to solve as the original Navier-Stokes equations but have the shortcoming that the theories are largely empirical and have limited regions of applicability.


Fig. 13. Just as Fig. 10 illustrates the various terms of the Navier-Stokes momentum equation, this figure illustrates the various terms of the Reynolds-stress transport equation. The driving force and the diffusion terms appear twice because each can be decomposed into a contribution to the transport of turbulence and a contribution to the generation or diffusion of turbulence.

In contrast to point-functional theories are the history-dependent, or turbulencetransport, theories. These theories, the focus of our interest here, include a set of one or more auxiliary equations that describe the history, or transport, of the variables associated with turbulence and that are solved in conjunction with the mean-flow equations (Eqs. 13). The auxiliary equations can range from empirical postulations to some form of the Reynolds transport equation (Eq. 15).

Because our starting point was the Navier-Stokes equations, turbulence-transport theory based on Eqs. 13 and 15 should, in principle, contain all the necessary information to describe the mean properties of turbulent flow. However, in practice it is necessary to introduce additional constraints or empirical information to yield a solvable set of equations. This procedure of "closing" the set of governing equations is called closure modeling and plays a central role in turbulence-transport theory.

The development of a solvable set of equations is beyond the scope of this article (although, in the following section we do so for a simple treatment of turbulence). We can nevertheless capture much of the flavor of the necessary developments by considering the significance of the terms in the Reynolds transport equation (Fig. 13 graphically illustrates the nature of each) and by considering the difficulties of describing their properties in terms of the macroscopically accessible mean-field quantities.

Advection, Mean-Flow Source, and Rotation. The advection and the mean-flow source and rotation terms of Eq. 15 contain only the unknown tensor R, and the meanflow velocities; no reference to the detailed turbulence structure occurs. These terms constitute a bulwark of settled mathematical structure for which there are essentially no uncertainties or controversies about the physics. In essence they describe the manner

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in which the mean flow moves turbulence from one place to another by translation, rotation, and stretching or contraction of the fluid.

Triple Correlation. The triple-correlation tensor uuu that appears in the next term in Eq. 15 is usually interpreted as a diffusive flux of the Reynolds stress generated by the action of the stress itself. Thus, this term can be called the turbulence self-diffusion term because it describes the turbulent diffusion of turbulence.

We can show in more detail how this identification is made and, at the same time, illustrate what is meant by closure modeling. If Q represents some quantity (such as the concentration of a dissolved, neutrally-buoyant substance) that is purely advected by the incompressible fluid, its transport equation is simply

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Decomposing the variables into mean and fluctuating parts and taking the ensemble average (as we did before with Eqs. 12 and 13), we find that

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Since the right side describes the diffusion of Q due to the effects of turbulence, we directly identify Q'u as a diffusive flux. Just as the flux of a chemical species is proportional to its concentration gradient (Fick's law), the diffusive flux is proportional to the gradient of Q itself:

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The proportionality constant is a function of the turbulence intensity; indeed, more detailed considerations indicate that

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In this manner, we see what is meant by closure modeling, that is, the elimination of any residual reference to details of the turbulence. For our purposes we need not delve any deeper into this aspect of turbulence modeling; the example is sufficient to indicate some of the heuristic and empirical procedures we inevitably have been forced to employ.

Driving Force. The pressure-velocity correlation terms (the first two terms on the right side of Eq. 15) are especially important to the transport modeling of turbulence. They describe one of the principal driving forces by which mean-flow energy finds its way

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